That set is basically the norm, but it's one of those sets where it doesn't necessarily work if you obtain it piece by piece. It works well in its completed form, but you'd lose out on one thing or another if you to say, use Thaumas Coat without having Koga Hakama +2 as well and things like that.

A more immediate setup for you might be something along the lines of the set up I listed above, but rather than Thaumas body, Koga +2 legs, and Iga +2 feet, stick with your Ninja Chainmail and use Iga Hakama +2 and something like Usukane Sune-ate as your feet. You'll lose out on a little bit of overall attack speed in comparison to the listed set, but it's a very good interim gear set till you can get some of the harder to obtain pieces to make the above set work.

And just so you know, you didn't link your gear set properly off AH. The link you wanted to use was this, not what you used, since that /members/ and id= redirects me to my own gear sets rather than yours.

Would I be able to go down the STR elemental path then again through the AGI path?

Also do you have any guesstimate on how long these take to complete from square 1? I know it depends on playstyle, but any kind of ballpark?

Yes.

Wind path you should be able to complete within a few days as the trials require wind weather or day, which wind weather is more frequent when it is spring ingame. Alot of people work on wind path trials so teaming up is also an option. Fire path will take you a few weeks because fire weather is very rare but still occurs around summer ingame for a short time, as you would be doing most of the trials during firesday.